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- Thread starter Shaybay92
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CompuChip

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[itex]|x - a| < \delta \implies |f(x) - L| < \epsilon[/itex] if and only if [itex]\delta < \delta_0[/itex]. But since the definition only states that there exists

Or were you asking: why is it possible that the delta we need depends on the point in which we are taking the limit? In that case, you might want to read up about different forms of continuity. For example, we say that a function is continuous at a, if

[tex]\forall \epsilon, \forall a, \exists \delta = \delta(a): |x - a| < \delta \implies |f(x) - f(a)| < \epsilon[/tex]

and then a function is continuous if it is continuous at all a (that is for all a and all epsilon we can find a delta which works for that epsilon and that a).

We can also require that

[tex]\forall \epsilon, \exists \delta, \forall a, |x - a| < \delta \implies |f(x) - f(a)| < \epsilon[/tex]

That is, for all epsilon we can find a single delta which works for that epsilon, but for

If such a delta

About the question about the asymptote, it is not quite clear to me what you mean here. Do you mean: why can't we find a delta around, for example, x = 0 for f(x) = 1/x?

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